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Mathematics

Higher Mini-Prelim Examination 2008/2009

Assessing Unit 3 + revision from Units 1 & 2

Time allowed - 1 hour 10 minutes

Read carefully 1. Calculators may be used in this paper. 2. Full credit will be given only where the solution contains appropriate working. 3. Answers obtained from readings from scale drawings will not receive any credit.

NATIONAL QUALIFICATIONS

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FORMULAE LIST Circle: The equation 02222 =++++ cfygxyx represents a circle centre ),( fg −− and radius cfg −+ 22 . The equation represents a circle centre ( a , b ) and radius r.

( )( )

AA

AAAAAA

BABABABABABA

2

2

22

sin211cos2sincos2cos

cossin22sinsinsincoscoscossincoscossinsin

−=

−=

−=

==±

±=±

Scalar Product: .andbetweenangletheisθwhere,cosθ. bababa = or

=ba .⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=++

3

2

1

3

2

1

332211

bbb

andaaa

where babababa

axaaxaxaax

xfxf

sincoscossin

)()(

−

ʹ′

Trigonometric formulae:

Table of standard derivatives:

Table of standard integrals:

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1. A is the point )5,6,4(− and B is the point )2,3,1(− . The components of AB are

A ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

333

B ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

795

C ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

333

D ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

795

2. The gradient of the tangent to the curve xy 2sin3= at the point where 6

π=x is

A 33

B 3

C 3−

D 33−

3. The circle 01071122 =++++ yxyx cuts the x-axis at the points P and Q.

The length of PQ is

A 3

B 7

C 9

D 11 4. Given that C is a constant of integration, then dxx 2

1)34( −

+ equals

A Cx ++ 21)34(

B Cx ++ 21)34(2

1

C Cx ++ 21)34(4

1

D Cx ++−−23

)34(2

SECTION A

In this section the correct answer to each question is given by one of the alternatives A, B, C or D. Indicate the correct answer by writing A, B, C or D opposite the number of the question on your answer paper. Rough working may be done on the paper provided. 2 marks will be given for each correct answer.

∫

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5. The derivative of 3)43( x− with respect to x is

A 16)43( 4x−

−

B 4)43( 4x−

C 4)43( x−−

D 2)43(12 x−−

6. Vector a has components a = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

k23

.

If 4=a , then the value of k is

A 3

B 1−

C 13−

D 3 7. Solve 3log3log 33 =+ xx , for x where 0>x .

A 1

B 427

C 3

D 43

8. The maximum value of 5cos4sin3 +− xx is

A 10

B 0

C 4

D 5−

[ END OF SECTION A ]

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9. Consider the diagram below. (a) Given that Q divides PR in the ratio 2:1 , find the coordinates of Q. 3 (b) Hence prove that angle SQR is a right angle. 4

10. Evaluate ∫ dxx 2)23(

6−

. 5

11. Solve the equation 2cos3sin =+ xx for 3600 ≤< x . 6 12. Find the coordinates of the point on the curve 2423 +−−= xxxy where the gradient of the tangent is 1 and 0<x . 4

SECTION B

ALL questions should be attempted

R(6,3,0)

Q

P(0,15,-3)

S(3,13,4)

0

1

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13. The diagram shows two vectors a and b where ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

204

a and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

036

b .

The angle between the vectors is θ .

(a) Show clearly that 5

4cos =θ . 3

(b) Hence, or otherwise, find the exact value of θ2cos . 2 14. The mass of radium-226 remaining after a decay period of t years can be

calculated using the formula

tkeMMt 0= , where 0M is the initial mass, tM is the mass remaining after t years and k is a constant.

(a) Find the value of the constant k, given that a sample of radium-226 takes 500 years to decay to 80% of its initial mass. Give your answer correct to 2 significant figures. 5 (b) Hence calculate the approximate percentage mass remaining, of a sample of

radium-226, after a period of 5 thousand years. Give your answer correct to the nearest percent. 2

[ END OF SECTION B ]

[ END OF QUESTION PAPER ]

a

b

θ